3.31 \(\int e^{\text {csch}^{-1}(a x)} \, dx\)

Optimal. Leaf size=24 \[ \frac {\log (x)}{a}+x e^{\text {csch}^{-1}(a x)}-\frac {\text {csch}^{-1}(a x)}{a} \]

[Out]

(1/a/x+(1+1/a^2/x^2)^(1/2))*x-arccsch(a*x)/a+ln(x)/a

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Rubi [A]  time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.29, number of steps used = 5, number of rules used = 5, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6331, 29, 242, 277, 215} \[ x \sqrt {\frac {1}{a^2 x^2}+1}+\frac {\log (x)}{a}-\frac {\text {csch}^{-1}(a x)}{a} \]

Warning: Unable to verify antiderivative.

[In]

Int[E^ArcCsch[a*x],x]

[Out]

Sqrt[1 + 1/(a^2*x^2)]*x - ArcCsch[a*x]/a + Log[x]/a

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 242

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^2, x], x, 1/x] /; FreeQ[{a, b, p},
x] && ILtQ[n, 0]

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 6331

Int[E^ArcCsch[(a_.)*(x_)^(p_.)], x_Symbol] :> Dist[1/a, Int[1/x^p, x], x] + Int[Sqrt[1 + 1/(a^2*x^(2*p))], x]
/; FreeQ[{a, p}, x]

Rubi steps

\begin {align*} \int e^{\text {csch}^{-1}(a x)} \, dx &=\frac {\int \frac {1}{x} \, dx}{a}+\int \sqrt {1+\frac {1}{a^2 x^2}} \, dx\\ &=\frac {\log (x)}{a}-\operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x^2}{a^2}}}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=\sqrt {1+\frac {1}{a^2 x^2}} x+\frac {\log (x)}{a}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^2}\\ &=\sqrt {1+\frac {1}{a^2 x^2}} x-\frac {\text {csch}^{-1}(a x)}{a}+\frac {\log (x)}{a}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 35, normalized size = 1.46 \[ \frac {a x \sqrt {\frac {1}{a^2 x^2}+1}+\log (a x)-\sinh ^{-1}\left (\frac {1}{a x}\right )}{a} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcCsch[a*x],x]

[Out]

(a*Sqrt[1 + 1/(a^2*x^2)]*x - ArcSinh[1/(a*x)] + Log[a*x])/a

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fricas [B]  time = 0.45, size = 86, normalized size = 3.58 \[ \frac {a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x + 1\right ) + \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x - 1\right ) + \log \relax (x)}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/a/x+(1+1/a^2/x^2)^(1/2),x, algorithm="fricas")

[Out]

(a*x*sqrt((a^2*x^2 + 1)/(a^2*x^2)) - log(a*x*sqrt((a^2*x^2 + 1)/(a^2*x^2)) - a*x + 1) + log(a*x*sqrt((a^2*x^2
+ 1)/(a^2*x^2)) - a*x - 1) + log(x))/a

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giac [A]  time = 0.13, size = 66, normalized size = 2.75 \[ -\frac {{\left (\log \left (\sqrt {a^{2} x^{2} + 1} + 1\right ) \mathrm {sgn}\relax (x) - \log \left (\sqrt {a^{2} x^{2} + 1} - 1\right ) \mathrm {sgn}\relax (x) - 2 \, \sqrt {a^{2} x^{2} + 1} \mathrm {sgn}\relax (x)\right )} {\left | a \right |}}{2 \, a^{2}} + \frac {\log \left ({\left | x \right |}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/a/x+(1+1/a^2/x^2)^(1/2),x, algorithm="giac")

[Out]

-1/2*(log(sqrt(a^2*x^2 + 1) + 1)*sgn(x) - log(sqrt(a^2*x^2 + 1) - 1)*sgn(x) - 2*sqrt(a^2*x^2 + 1)*sgn(x))*abs(
a)/a^2 + log(abs(x))/a

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maple [B]  time = 0.05, size = 113, normalized size = 4.71 \[ -\frac {\sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, x \left (-\sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2}+\ln \left (\frac {2 \sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2}+2}{a^{2} x}\right )\right )}{\sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2}}+\frac {\ln \relax (x )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/a/x+(1+1/x^2/a^2)^(1/2),x)

[Out]

-((a^2*x^2+1)/a^2/x^2)^(1/2)*x*(-(1/a^2)^(1/2)*((a^2*x^2+1)/a^2)^(1/2)*a^2+ln(2*((1/a^2)^(1/2)*((a^2*x^2+1)/a^
2)^(1/2)*a^2+1)/a^2/x))/(1/a^2)^(1/2)/((a^2*x^2+1)/a^2)^(1/2)/a^2+ln(x)/a

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maxima [A]  time = 0.31, size = 64, normalized size = 2.67 \[ x \sqrt {\frac {1}{a^{2} x^{2}} + 1} - \frac {\log \left (a x \sqrt {\frac {1}{a^{2} x^{2}} + 1} + 1\right )}{2 \, a} + \frac {\log \left (a x \sqrt {\frac {1}{a^{2} x^{2}} + 1} - 1\right )}{2 \, a} + \frac {\log \relax (x)}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/a/x+(1+1/a^2/x^2)^(1/2),x, algorithm="maxima")

[Out]

x*sqrt(1/(a^2*x^2) + 1) - 1/2*log(a*x*sqrt(1/(a^2*x^2) + 1) + 1)/a + 1/2*log(a*x*sqrt(1/(a^2*x^2) + 1) - 1)/a
+ log(x)/a

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mupad [B]  time = 2.25, size = 36, normalized size = 1.50 \[ \frac {\ln \relax (x)}{a}+x\,\sqrt {\frac {1}{a^2\,x^2}+1}+\frac {\mathrm {asin}\left (\frac {1{}\mathrm {i}}{a\,x}\right )\,1{}\mathrm {i}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/(a^2*x^2) + 1)^(1/2) + 1/(a*x),x)

[Out]

log(x)/a + (asin(1i/(a*x))*1i)/a + x*(1/(a^2*x^2) + 1)^(1/2)

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sympy [A]  time = 1.14, size = 48, normalized size = 2.00 \[ \frac {x}{\sqrt {1 + \frac {1}{a^{2} x^{2}}}} + \frac {\log {\relax (x )}}{a} - \frac {\operatorname {asinh}{\left (\frac {1}{a x} \right )}}{a} + \frac {1}{a^{2} x \sqrt {1 + \frac {1}{a^{2} x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/a/x+(1+1/a**2/x**2)**(1/2),x)

[Out]

x/sqrt(1 + 1/(a**2*x**2)) + log(x)/a - asinh(1/(a*x))/a + 1/(a**2*x*sqrt(1 + 1/(a**2*x**2)))

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